Hexagonal two-dimensional lattice

The Ir(100)(lx5) reconstruction is caused by a lateral distortion of the top layer of Ir atoms along the (10) direction (Lang et al., 1983). This distortion allows the top layer of Fr atoms to form a quasi-hexagonal two-dimensional lattice which is commensurate with the underlying (100) plane formed by the... 

Discotic liquid crystals arise from disk-shaped molecules as nematic or cholesteric mesophases. Their structural characteristics are similar to the respective ealamitie mesophases, that is, the normals of the disks are oriented parallel. Instead of the smectic mesophases, diseotie columnar liquid crystals arise from eonnecting the disks to each other. The columns of the discotic columnar mesophase form a two-dimensional lattice whieh is in a hexagonal or rectangular modification. In addition, the columns may be tilted (Fig. 2f,g). 

Note Depending on the order in the molecular stacking in the columns and the two-dimensional lattice symmetry of the columnar packing, the columnar mesophases may be classified into three major classes hexagonal, rectangular and oblique (see Definitions 3.2.2.1. to 3.2.2.3). 

The critical value, C0, is larger when C > 1 than it is for the case C = 1, as may be verified by direct calculation or by comparison of the size of the loops in Figure 3. This again is an expected result for it is known that C0 as determined from the Fowler-Guggenheim approximation is too low. Here, the exact solutions serve as a comparison standard by sophisticated mathematical techniques reviewed in several articles (2, 7) it is possible to determine the exact value of C0 for certain types of infinite two-dimensional lattices, the square and hexagonal arrays among them. Values for C0 in the present approximation were obtained by trial and error techniques. The results are presented in Table IV. 

Separation. Consider a hypothetical plane parallel to the surface of the cast solution. Assume that the secondary particles are placed on this plane In a two-dimensional lattice of hexagonal close packing (see Figure 4). The lattice coordination number Is... 

Besides smectic mesophases, columnar mesophases generally present a more pronounced order, since they result from the self-assembly of columns of indefinite length according to a two-dimensional lattice [2]. The symmetry of the latter is found to be hexagonal, rectangular or oblique (Fig. 1) [3]. 

Fig. la-e. Different types of lattices found in columnar mesophases. Two-dimensional lattice with a) hexagonal symmetry, b) rectangular symmetry (P2i/a), c) rectangular symmetry (P2/ a), d) centred rectangular symmetry (C2/m), e) oblique (PI). (From Levelut AM (1983) J Chimie Physique 80 149)... 

Such molecules exhibit columnar mesophases and smectic C phases. In contrast to the corresponding tetracatenar compounds described above, the columnar phase is, in most of these swallow-tailed compounds, the low temperature phase with respect to the smectic C phase [58]. This inversion of thermotropic sequence with respect to the biforked compounds described just above has not found any explanation in relation with molecular parameters. The symmetry of the columnar two-dimensional lattice is found to be hexagonal, oblique and even rectangular [59] in some cases. For the latter case, the model proposed involves several molecules packed side by side to form the lattice unit, the different moieties being packed alternately in order to form a two-dimensional centred cell as shown in Fig. 11. 

Fig. 2.38 A two-dimensional lattice of A atoms (a) with hexagonal potentials the primitive unit cells are replaced by overlapping circles (b) a muffin-tin potential (c) for the hexagonal lattice, composed of circular potentials (atomic regions) with flat potentials (interstitial regions) inbetween.

The first liquid crystals of disc-shaped molecules, now generally referred to as discotic liquid crystals, were prepared and identified in 1977. Since then a large number of discotic compounds have been synthesized and a variety of mesophases discovered. Structurally, most of them fall into two distinct categories, the columnar and the nematic. The columnar phase in its simplest form consists of discs stacked one on top of the other aperiodically to form liquid-like columns, the different columns constituting a two-dimensional lattice (fig. 1.1.8 (a)). The structure is somewhat similar to that of the hexagonal phase of soap-water and other lyotropic... 

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